Finite Group Databasebeta

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Cayley diagram

C4×#951 #82532

order384
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C25

Aliases

direct productC4×#951

Splittings

SylowC3 ⋊ C4×#137

Subgroups

subgroupG/H
centerC2×C4C2×#67
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×C4×#41C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
cosocleC2×C3×C4C24
hypercenterC4×#41D3
hypocenterC3C4×#137
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#41

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(9,14)(10,17)(11,19)(13,20)(15,16)(21,22);(8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,21)(19,23);(2,3)(8,9,12,16)(10,13,17,21)(11,14,19,15)(18,22,23,20);(4,5,6,7)

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10689802, #10689804, #10689816, …
maximal normal subgroup of#10689802, #10689804, #10689816, …
maximal quotient ofC8×#951, #10764779, #10764805, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 15108);